arXiv subjects

Tomoyoshi Ibukiyama

Publications and source records attributed to Tomoyoshi Ibukiyama.

11 recordsLinked to original sources

arXiv2026

Harder's conjecture II

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $P$ be a prime ideal of the Hecke field of $f$. We denote by $Sp_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k$ is congruent to $0$ modulo $4$, $j$ is congruent to $0$ modulo $4$, and that $P$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform $F$ in the space of modular forms of weight $(k+j,k)$ for $Sp_2(Z)$ such that $[I_2(f)]^{\bf k}$ is congruent to $A^{(I)}_4(F)$ modulo $P$. Here, $[I_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $I_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $Sp_4(Z)$, and $A^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $Sp_4(Z)$. As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of $F$ and some quantities related to the Hecke eigenvalues of $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.

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arXiv2026

Modules of Jacobi forms of degree two of small levels

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the full Siegel modular group as a module over scalar valued Siegel modular forms of even weight has been explicitly given by T.~Ibukiyama. There we used an explicit structure theorem of rings of scalar valued Siegel modular forms by Igusa and that of vector valued Siegel modular forms of weight $\det^k\, \Sym^2$ by T. Satoh and T. Ibukiyama. On the other hand, for levels $N=2$, $3$, $4$, ring structures of scalar valued case have been also known by H. Aoki and T. Ibukiyama and $\det^k \Sym^2$-valued case by H. Aoki. In this paper, by merging these results, we give the same sort of simple structure theorems on modules of Jacobi forms of degree of two of index one for level $2$, $3$ and $4$.

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arXiv2025

Hermitian modular forms and algebraic modular forms on $SO(6)$

We state conjectures that relate Hermitian modular forms of degree two and algebraic modular forms for the compact group $SO(6)$. We provide evidence for these conjectures in the form of dimension formulas and explicit computations of eigenforms.

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arXiv2025

When is a polarised abelian variety determined by its $\boldsymbol{p}$-divisible group?

We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.

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arXiv2024

Formal series of Jacobi forms

We prove for general paramodular level that formal series of scalar Jacobi forms with an involution condition necessarily converge and are therefore the Fourier-Jacobi expansions at the standard 1-cusp of paramodular Fricke eigenforms.

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arXiv2024

Dimensions of paramodular forms and compact twist modular forms with involutions

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$, as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornaría on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When $p$ is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small $p$ and $k$, and the complete list of primes $p$ such that there is no paramodular cusp form of level $p$ of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with $(1,p)$ polarization.

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arXiv2023

Period of the Ikeda-Miyawaki lift

In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.

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arXiv2023

Genus character $L$-functions of quadratic orders in an adelic way and maximal orders of matrix algebras

For a quadratic extension $K$ of ${\mathbb Q}$, we consider orders $O$ in $K$ that are not necessarily maximal and the ideal class group $Cl^+(O)$ in the narrow sense of proper ideals of $O$. Characters of $Cl^+(O)$ of order at most two are traditionally called genus characters. Explicit description of such characters is known classically, but explicit $L$-functions associated to those characters are only recently obtained partially by Chinta and Offen and completely by Kaneko and Mizuno. As remarked in the latter paper, the present author also obtained the formula of such $L$-functions independently. Indeed, here we will give a simple and transparent alternative proof of the formula by rewriting explicit genus characters and their values in an adelic way starting from scratch. We also add an explicit formula for the genus number in the wide sense, which is maybe known but rarely treated. As an appendix we give an ideal-theoretic characterization of isomorphism classes of maximal orders of the matrix algebras $M_n(F)$ over a number field $F$ up to $GL_n(F)$ and $GL_n^+(F)$ conjugation respectively, and apply genus numbers to count them when $n=2$ and $F$ is quadratic. To avoid any misconception, we include some easy known details.

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arXiv2022

Differentiating Siegel modular forms, and the moving slope of ${\mathcal A}_g$

We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal A}_g$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal A}_5$ and a conjectural upper bound for the moving slope of ${\mathcal A}_6$.

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arXiv2022

Harder's conjecture I

Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.

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arXiv2012

Jacobi forms that characterize paramodular forms

The Fourier Jacobi expansions of paramodular forms are characterized from among all formal series of Jacobi forms by two conditions on the Fourier coeffcients of the Jacobi forms: a growth condition and a set of linear relations. Examples, both theoretical and computational, indicate that the growth condition may be superfluous.

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